PURPOSE

To study wave motion and its superposition as exemplified by standing (stationary) waves on a vibrating string.

EQUIPMENT  tuning fork oscillator, mass holder and assorted masses, meter stick, thin paper strips.

RELEVANT EQUATIONS

Wavelength-Frequency relation		λ = v / f   or   v = f λ
Wave Speed on a String
Distance between Adjacent Nodes		LN = λ/2

DISCUSSION

The frequency of a wave motion is determined by the source of vibration. The wavelength depends on the speed of the wave, which is determined by the medium of propagation. Thus, for any wave:

λ = v / f

(1)

where λ is the wavelength (meters in SI units); v is the wave velocity (m/s); and f is the frequency (hertz). For transverse waves in a stretched cord having a stretching force or tension T, it can be shown that:

(2)

where v is the wave velocity along the cord (m/s); T is the tension (newtons); and, μ is the linear mass density of the string = (kg/m).

When two waves of equal amplitude are propagated along a string in opposite directions, a special pattern of vibration is produced. This pattern appears to the naked eye to be standing still, hence the name "standing wave". Actually, the string vibrates rapidly in loops, which are separated by completely still points called Nodes. The middle of the vibrating loops are called Antinodes. The standing wave pattern results from the spatial periodicity of the propagating waves. The distance between adjacent Nodes is one-half of a wavelength, λ/2. By setting up a standing wave pattern, we thus have a simple method for determining the wavelength of the propagating waves. This technique can be used with any type of wave motion.

The apparatus for this experiment will allow you to set up transverse standing mechanical waves on a string. By eliminating v between Equations (1) and (2), we obtain a relation between quantities that are readily measurable: wavelength, frequency, and tension. The linear mass density μ, is a property specific to the string being used.

(3)